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Dynamic Modeling and Simulation of an AutonomousUnderwater Vehicle (AUV)A ThesisSUBMITTED TO THE FACULTY OF THEUNIVERSITY OF MINNESOTABYKevin OrpenIN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR THE DEGREE OFBachelor of Aerospace Engineering and MechanicsWith HonorsFaculty Adviser: Professor Junaed Sattar, PhDMay 2021

Copyright PageCopyright 2021 by Kevin Orpen

AcknowledgementsThis thesis, and my overall experience with the world of underwater robotics, would nothave been possible without my adviser, Professor Junaed Sattar. I want to thank you forwelcoming me into the Interactive Robotics and Vision Laboratory with open arms. Youhave always been such a strong advocate of mine and supported my learning throughoutthese past years, and for that, I will forever be grateful.I would also like to thank Professors Maziar Hemati and Yohannes Ketema for serving onmy thesis committee. Your support and dedication to my research has greatly helped merefine my work and make this final product a reality.i

AbstractAutonomous Underwater Vehicles (AUVs) have been in development in recent decades toaddress the difficulties and high costs of oceanic exploration, with applications includingmarine life monitoring, search and rescue operations, and wreck inspection. An underwaterrobot developed by the Interactive Robotics and Vision (IRV) Laboratory at the Universityof Minnesota is LoCO, a Low Cost Open-Source AUV. LoCO seeks to assist in a numberof underwater applications while reducing the current high cost of entry into underwaterrobotics. One aspect of this underwater vehicle that is integral to its capacity as an AUV isthe modeling of its dynamics, and each new AUV comes with unique geometries spanningvarious propulsion control methods for specializing in different underwater tasks. Thisthesis seeks to establish an underwater dynamic model for the robot, implement the modelin a simulated setting so as to provide testing opportunities before field deployment, andcompare the effectivity of the model to collected experimental data. This, in turn, will leadto the efficient development of its autonomous systems and capability to assist inunderwater operations. Within this research, the dynamic models have been produced andgeometry-dependent coefficients have been derived for LoCO. A simulator for the robothas also been developed that can interface with onboard software. Though the simulationagrees relatively well with experimental data collected for LoCO’s forward motion, thereare still other motion modes that require further investigation. Overall, this dynamicfoundation will provide for future control system and other autonomous development tofurther its underwater capabilities.ii

Table of ContentsList of Tables . viList of Figures . viiChapter 1Introduction . 11.1.Background . 11.2.LoCO Design. 21.3.Thesis Objective . 31.4.Overview . 4Chapter 2Derivation of Dynamic Equations . 52.1.Introduction . 52.2.Body Frame Definition. 52.3.Relationship Between Inertial and Body Coordinate Frames . 82.4.Rigid Body 6-Degrees of Freedom Equations of Motion . 102.5.Forces and Moments . 122.5.1.Environmental Forces . 122.5.2.Propulsion . 132.5.3.Restoring Forces . 142.5.4.Added Mass . 14iii

2.5.5.Hydrodynamic Damping . 172.6.Overall Kinetics Equations. 192.7.Conclusion. 20Chapter 3Estimation of LoCO Dynamic Parameters . 213.1.Introduction . 213.2.Buoyancy . 213.3.Mass and Moments of Inertia . 233.4.Added Mass Coefficients . 263.5.Hydrodynamic Damping Coefficients . 363.6.Dynamic Equations for LoCO. 423.7.Conclusion. 44Chapter 4Simulation . 464.1.Introduction . 464.2.Modeling and Visualization . 464.3.Physics. 484.4.Simulation Architecture and Control . 494.5.Quaternions and Force Application. 504.6.Conclusion. 51Chapter 5Experimental Data and Comparison with Simulation . 53iv

5.1.Introduction . 535.2.Experiment and Data Analysis . 535.3.Simulation Data Comparison . 555.4.Conclusion. 57Chapter 6Conclusion . 586.1.Review. 586.2.Conclusions . 586.3.Future Work . 59References . 60Appendix A . 63v

List of TablesTable 1: Vehicle buoyancy analysis summary. 22Table 2: Vehicle mass and moments of inertia analysis summary. . 25Table 3: Table of components and corresponding variable names for geometric LoCOmodels. . 29Table 4: Added mass analysis values. 36Table 5: Summary of component coefficients of drag. 38Table 6: Summary of component drag reference areas. . 38Table 7: Summary of hydrodynamic damping analysis results. . 42Table 8: Relevant dynamic model parameters from analysis and estimation. . 43Table 9: LoCO Component Matrix for mass and moment of inertia determination. . 64vi

List of FiguresFigure 1: LoCO in untethered deployment in the Caribbean Sea near Barbados [4]. . 2Figure 2: LoCO CAD model in SolidWorks. . 2Figure 3: LoCO body-fixed coordinate frame definition overall view. . 6Figure 4: LoCO body-fixed coordinate frame definition section views. . 7Figure 5: Relationship between body frame and inertial frame. . 8Figure 6: Example of notation for rectangular prism of square cross section rotatingabout one end (rotation about the dashed line). . 18Figure 7: Moments of inertia for hollow cylinder and rectangular prism [20]. . 24Figure 8: Parallel axis theorem example, where m is mass of the object. . 25Figure 9: Geometric approximation for LoCO flow in x-direction – “X-Model”. . 28Figure 10: Geometric approximation for LoCO flow in y-direction – “Y-Model”. . 28Figure 11: Geometric approximation for LoCO flow in z-direction – “Z-Model”. 29Figure 12: Added mass parameters for a circle and a square [13], [21]. . 30Figure 13: LoCO in a simulated Gazebo world. . 47Figure 14: ROS program nodes and messages used to run the simulation. . 50Figure 15: Thruster force versus PWM input [28]. 54Figure 16: Velocity versus forward thrust applied for a straight line, horizontal path in apool. . 55Figure 17: Velocity versus full forward thrust applied for a straight line, horizontal path. 56vii

Figure 18: Clockwise penetrator numbering convention. . 64viii

Chapter 1Introduction1.1.BackgroundOcean exploration began in the late 19th century with the search for a greaterunderstanding of how the Earth works and the life it holds. Since then, discoveries acrossthe span of 71% of the Earth’s surface have yielded entirely new fields of study andrevolutionary findings in areas such as geology, marine biology, and environmentalscience. However, the majority of Earth’s oceans remain unexplored, due in part to highresource and economic costs [1]. This, along with the quest to explore deep sea areasextremely difficult for humans to reach, has spurred the field of underwater robotics.Autonomous Underwater Vehicles (AUVs) have been in development in recent decadesto address the high cost of underwater exploration. Other applications of thesetechnologies include marine life monitoring, mineral exploration, global environmentevaluation, wreck inspection, and search and rescue operations. The dynamic underwaterenvironment comes with a host of new challenges though, ranging from navigation whereGPS does not function to environments of operation prone to unpredictable disturbances[2]. An example of underwater robotics development for oceanography research can befound with the Seaglider [3], where the AUV is designed to operate for long periods oftime to gather ocean data at a fraction of the cost of manned expeditions. Thoughresearch in underwater robotics has greatly progressed, the sensors and equipmentrequired for accurate navigation and reliable operation in underwater environments oftencome at high costs.1

1.2.LoCO DesignAn AUV developed by the Interactive Robotics and Vision (IRV) Laboratory atthe University of Minnesota is LoCO. LoCO AUV is a Low-Cost, Open-Source,Autonomous Underwater Vehicle. It is vision-guided and rated to a depth of 100 meters,capable of being deployed by a single person in the field [4]. A picture of LoCO in anocean deployment can be seen below, along with the corresponding Computer AidedDesign (CAD) model in SolidWorks. This underwater vehicle seeks to assist personnel innumerous underwater applications in a more cost-effective manner than previous robots.Figure 1: LoCO in untethered deployment in the Caribbean Sea near Barbados [4].Figure 2: LoCO CAD model in SolidWorks.2

One aspect of LoCO that is integral to its capacity as an AUV is the modeling ofits dynamics. For a reliable control system to be developed that enables autonomousoperation, a foundation in the dynamics of the robot must be established. Though anumber of studies have been conducted on the modeling of AUVs in an underwaterenvironment [5], [6], and the governing equations for marine hydrodynamics have beenestablished, each new AUV comes with unique geometries. Further, AUV designs boastvarious propulsion control methods for the underwater tasks they are primarily designedfor.1.3.Thesis ObjectiveThis thesis seeks to evaluate the known properties common to any underwaterenvironment and apply foundational components to characterize the underwaterdynamics of the LoCO AUV. Work is performed to implement these dynamics in asimulated environment, able to be integrated with onboard robot systems, in order toprovide a mode of testing software before resource-costly field trials. Experimentaltesting data is compared to the simulation implementation of the dynamic model. Theseoutcomes, in turn, will lead to the efficient development of future control systems toassist in underwater research including trash detection development, vision systemdevelopment, and human-robot interaction research [7].3

1.4.OverviewThe content discussed herein serves to first provide a foundation in the governingdynamics and various definitions for LoCO in Chapter 2. A number of assumptions arepreviewed before a brief description of the relation between dynamics derived in theframe of a rigid body and the corresponding motion in an inertial frame. The generaldynamic equations for a 6-Degrees-of-Freedom (DoF) body are derived and associatedforces for the underwater robot are discussed. Chapter 3 takes these general equations anddescribes the various estimation methods used to evaluate the coefficient parameters inthe dynamic equations specific to the vehicle. A final list of estimated parameters ispresented along with the resulting simplified dynamic equations. Chapter 4 discusses thesimulation component of the thesis, including its program architecture and application offorces. Chapter 5 looks at how the simulation is utilized to compare the dynamic model toexperimental data collected. Finally, Chapter 6 summarizes the thesis in conclusion of thework.4

Chapter 2Derivation of Dynamic Equations2.1.IntroductionBefore any specific dynamic estimates regarding LoCO can be made, the dynamicframework and general equations of motion must be derived. Though these derivationscan be found in a number of texts [8]–[10], they are included within this thesis so that thedocument may be self-sufficient. Governing assumptions that are being made for thisdynamic model will be explained in each of the appropriate sections, but are listed belowas an overview [8]:1) The AUV can be treated as a rigid body of a constant mass.2) The earth’s rotation is negligible for acceleration components of the vehicle’scenter of mass.3) The thrusters are assumed to be a constant source of thrust.4) The underwater vehicle is sufficiently submerged in an unbounded and ideal fluid.5) The AUV does not experience underwater currents.6) The AUV is assumed to not be a streamlined body due to external irregularitiessuch as clamps.2.2.Body Frame DefinitionThere are often two frames of reference used in expressing the equations ofmotion for an underwater vehicle. One of these is a “body” coordinate frame, which isfixed to the body of the marine vehicle. It is beneficial in many other applications to5

attach the origin of the coordinate system to the center of gravity of the vehicle.However, especially with a vehicle that can be modified such as LoCO, the center ofgravity may change throughout the design cycle of the vehicle. Along with modelingbenefits explained later regarding body-fixed origin placement along lines of symmetry,it is common to define the body-fixed coordinate frame origin at a location other than thecenter of gravity. The coordinate system for LoCO can be seen below in Fig. 3. In marineengineering, velocities in the x, y, and z directions are defined as surge (u), sway (v), andheave (w), respectively, with overall forces along the axes noted as X, Y, and Z.Similarly, angular velocities about those same axes are defined as roll (p), pitch (q), andyaw (r), respectively, with overall moments along the axes noted as K, M, and N.Figure 3: LoCO body-fixed coordinate frame definition overall view.6

Figure 4: LoCO body-fixed coordinate frame definition section views.As can be seen in Figs. 3 and 4, LoCO has geometric planes of symmetry aboutthe x-z plane and x-y plane. For the case of this analysis, the body frame origin for LoCOis located at the intersection of these two planes and along the x axis to the back face ofthe rear end cap. This way, the characteristics of the robot can change with minimaleffect to the dynamic framework of the thesis. More will be discussed on the forcerelated advantages to this origin location as well.7

2.3.Relationship Between Inertial and Body Coordinate FramesThe second frame used as reference for these equations of motion is an inertialframe that the underwater vehicle operates in. For this analysis, an Earth-fixed inertialsystem will be used called “NED”, or “North-East-Down” [10]. In this case, the xdirection of the coordinate system points North, the y direction points East, and the zdirection therefore points downwards.Though much of this thesis is devoted to the analysis of motion and forces in thebody-fixed frame defined above, for any mission with an underwater vehicle, it is criticalto relate the state of the vehicle back to the overall frame of reference. A visual depictionof the relation between the Earth-fixed frame and the body-fixed frame is given below inFig. 5. The position of the vehicle in relation to the Earth frame is often given in terms ofXE, YE, and ZE, along their respective axes. The angle of rotation about each of these axesis denoted as ϕ, θ, and ψ, respectively.Figure 5: Relationship between body frame and inertial frame.8

Euler angles give a way of relating the orientation of a rigid body in threedimensional space with reference to an original reference frame through three single-axisrotations. Though there are multiple possible conventions for achieving this, one exampleis the Euler “3-2-1”, or “Z-Y-X”, sequence. Each rotation is represented by a 3 x 3matrix, with the first rotation about the original z-axis by the yaw angle, denoted asRz(ψ). Then, a rotation about the following y-axis by the pitch angle is performed, andfinally a rotation about the following x-axis by the roll angle. More detail on theserotation matrices and Euler angles can be found in various sources such as [11], [12].Overall, a final relationship between the velocity of the underwater vehicle in Earth framecoordinates and body frame coordinates can then be represented as𝑋𝐸̇𝑢[ 𝑣 ] 𝑅𝑥 (𝜙)𝑅𝑦 (θ)𝑅𝑧 (ψ) [ 𝑌𝐸̇ ]𝑤𝑍𝐸̇1 [0000cos (𝜃)cos (𝜙) sin (𝜙) ] [ 0 sin (𝜙) cos (𝜙) sin (𝜃)0 sin (𝜃) cos (𝜓) sin (𝜓) 0 𝑋𝐸̇10] [ sin (𝜓) cos (𝜓) 0] [ 𝑌𝐸̇ ]0 cos (𝜃)001 𝑍𝐸̇cos (𝜃)cos (𝜓) [ sin(𝜓) cos(𝜙) sin (𝜙)sin (𝜃)cos (𝜓) sin(𝜙) sin(𝜓) cos (𝜙)cos (𝜓)sin (𝜃)cos (𝜃)sin (𝜓)cos(𝜙) cos(𝜓) sin (𝜙)sin (𝜃)sin (𝜓) sin(𝜙) cos(𝜓) cos (𝜙)sin (𝜃)sin (𝜓)𝑋𝐸̇ sin (𝜃)sin (𝜙)cos (𝜃) ] [ 𝑌𝐸̇ ]cos (𝜙)cos (𝜃) 𝑍𝐸̇(1)9

To achieve a similar transformation for angular velocities, each Euler anglederivative must be addressed separately since they are not independent orthogonalelements. So,0𝑝0𝜙̇[𝑞 ] 𝑅𝑥 (ϕ)𝑅𝑦 (θ)𝑅𝑧 (ψ) [ 0 ] 𝑅𝑥 (ϕ)𝑅𝑦 (θ) [𝜃̇] 𝑅𝑥 (ϕ) [ 0 ]𝑟𝜓̇00𝑝 𝜙̇ sin (𝜃)𝜓̇(2a)𝑞 sin(𝜙) cos(𝜃) 𝜓̇ 𝑐𝑜𝑠(𝜙)𝜃̇(2b)𝑟 cos (𝜙)cos (𝜃)𝜓̇ sin(𝜙)𝜃̇(2c)One drawback of Euler angle representation of the motion of the vehicle is what isknown as “gimbal lock”, or when there is a singularity in the equations of motion, such asthat caused by a pitch value of 90 degrees. For the sake of expressing the relationships asmost commonly used in marine dynamics, the resulting equations of motion as derivedabove hold their value since large pitch angles are usually avoided with submarines orother streamlined underwater vehicles [9]. An alternate way to express rotations arequaternions. Though they can be more complicated than expressions with Euler angles,they eliminate the issue with singularity and provide computationally easier ways to workwith rotations. Quaternions, and how they are implemented in simulation, are discussedlater.2.4.Rigid Body 6-Degrees of Freedom Equations of MotionAs given in the first two assumptions, the equations of motion for LoCO can bederived as those for a rigid body with constant mass and six degrees of freedom. The10

following set of equations applies to a body-fixed coordinate frame in which the center ofgravity is not necessarily the center of the coordinate system, derived within NewtonEuler framework. This can be seen defined in Figs. 3 and 4 for LoCO, where the forcesand moments of the following derivation are taken about the origin of the body system.Beginning with an expression of Newton’s second law, the forces on the body areequivalent to the time derivative of linear momentum. When dealing with rotatingcoordinate frames inside an overall inertial frame, rotational effects on changes inmomentum must be compensated for besides linear acceleration. This leads to𝑑𝑭 𝑑𝑡 (𝑚𝑼) 𝑚 (𝑼̇ 𝝎 𝑼 𝜶 𝒓𝑮 𝝎 (𝝎 𝒓𝑮 ))(3)where bold notation indicates vectors, in which U is linear velocity of the vehicle origin,ω is the angular velocity of the body system, α is the angular acceleration, and rG is theposition vector from the origin of the coordinate frame to the vehicle center of gravity. Inthis equation, the first term in the translational acceleration, the second is the Coriolisterm, the third is the azimuthal acceleration, and the final term is the centripetalacceleration.Now for the moments on the vehicle, the sum of these is equivalent to the timederivative of angular momentum. Similar to dealing with a rotating frame in equation 3,𝑑𝑴 𝑑𝑡 (𝑰𝝎) 𝑰 𝜶 𝝎 (𝑰 𝝎) 𝑚 𝒓𝑮 (𝑼̇ 𝝎 𝑼)(4)where the same notation is used from equation 3, and I is the moment of inertia matrixfor the body given with the matrix,𝐼𝑥𝑰 [ 𝐼𝑦𝑥 𝐼𝑧𝑥 𝐼𝑥𝑦𝐼𝑦 𝐼𝑧𝑦 𝐼𝑥𝑧 𝐼𝑦𝑧 ]𝐼𝑧(5)11

More in-depth derivations of these equations of motion can be found in sourcessuch as [8]–[10]. Evaluating equation 3 for the axial force, lateral force, and vertical forceequations,𝑋 𝑚(𝑢̇ 𝑣𝑟 𝑤𝑞 𝑥𝐺 (𝑞 2 𝑟 2 ) 𝑦𝐺 (𝑝𝑞 𝑟̇ ) 𝑧𝐺 (𝑝𝑟 𝑞̇ ))(6a)𝑌 𝑚(𝑣̇ 𝑤𝑝 𝑢𝑟 𝑥𝐺 (𝑞𝑝 𝑟̇ ) 𝑦𝐺 (𝑟 2 𝑝2 ) 𝑧𝐺 (𝑞𝑟 𝑝̇ ))(6b)𝑍 𝑚(𝑤̇ 𝑢𝑞 𝑣𝑝 𝑥𝐺 (𝑟𝑝 𝑞̇ ) 𝑦𝐺 (𝑟𝑞 𝑝̇ ) 𝑧𝐺 (𝑝2 𝑞 2 ))(6c)Evaluating equation 4 for the moment about the roll moment, pitch moment, andyaw moment,𝐾 𝐼𝑥 𝑝̇ (𝐼𝑧 𝐼𝑦 )𝑞𝑟 (𝑟̇ 𝑝𝑞)𝐼𝑥𝑧 (𝑟 2 𝑞 2 )𝐼𝑦𝑧 (𝑝𝑟 𝑞̇ )𝐼𝑥𝑦 𝑚(𝑦𝐺 (𝑤̇ 𝑢𝑞 𝑣𝑝) 𝑧𝐺 (𝑣̇ 𝑤𝑝 𝑢𝑟))(6d)𝑀 𝐼𝑦 𝑞̇ (𝐼𝑥 𝐼𝑧 )𝑟𝑝 (𝑝̇ 𝑞𝑟)𝐼𝑥𝑦 (𝑝2 𝑟 2 )𝐼𝑥𝑧 (𝑞𝑝 𝑟̇ )𝐼𝑦𝑧 𝑚(𝑧𝐺 (𝑢̇ 𝑣𝑟 𝑤𝑞) 𝑥𝐺 (𝑤̇ 𝑢𝑞 𝑣𝑝))(6e)𝑁 𝐼𝑧 𝑟̇ (𝐼𝑦 𝐼𝑥 )𝑝𝑞 (𝑞̇ 𝑟𝑝)𝐼𝑦𝑧 (𝑞 2 𝑝2 )𝐼𝑥𝑦 (𝑟𝑞 𝑝̇ )𝐼𝑥𝑧 𝑚(𝑥𝐺 (𝑣̇ 𝑤𝑝 𝑢𝑟) 𝑦𝐺 (𝑢̇ 𝑣𝑟 𝑤𝑞))2.5.(6f)Forces and Moments2.5.1. Environmental ForcesOne of the primary assumptions for this dynamic analysis lies with theenvironmental forces. In a full seakeeping analysis of an underwater vehicle, surfaceeffects, radiation-induced damping, and other wave effects are taken into account. Thesevariables can come to be very dependent upon the situation involved for the vehicle.Since LoCO is designed to operate in a range of environments, these wave-dependent12

effects and surface effects are ignored with the assumption that the AUV is sufficientlysubmerged underwater.2.5.2. PropulsionThe propulsion for LoCO is governed with three thrusters. The rear port andstarboard thrusters control movement in the horizontal plane, while the vertical thrusteradds control in the vertical plane. Each thruster is referred to as “port”, “stbd”, and“fore”, respectively. One assumption regarding the analysis with the thrusters is that theyact as point forces at the center of the thruster propellers. Though the actual thrustershave inherent propeller dynamics, the point force simplification has been made for thisanalysis as a reasonable approximation. As for the reaction torques of each thruster, theport and starboard thrusters have propellers that provide the same forward thrust whilespinning in opposite directions, so the moments cancel each other. It also assumed thatthe moment produced by the force thruster is negligible with respect to the larger vehicledynamics and is ignored. With this, the forces and moments created by the thrusters canbe expressed with the same conventions as established in section 2.4.𝑋𝑃 𝑇𝑝𝑜𝑟𝑡 𝑇𝑠𝑡𝑏𝑑(7a)𝑌𝑃 0(7b)𝑍𝑃 𝑇𝑓𝑜𝑟𝑒(7c)𝐾𝑃 0(7d)𝑀𝑃 𝑇𝑓𝑜𝑟𝑒 𝑥𝑓𝑜𝑟𝑒(7e)𝑁𝑃 𝑇𝑝𝑜𝑟𝑡 𝑦𝑝𝑜𝑟𝑡 𝑇𝑠𝑡𝑏𝑑 𝑦𝑠𝑡𝑏𝑑(7f)13

2.5.3. Restoring ForcesThe gravity and buoyancy forces acting on an underwater vehicle are known asthe restoring forces. To create a stable underwater vehicle, it is generally desired that thecenter of buoyancy and center of gravity are located at the same x and y positions withrespect to the origin, and with the center of gravity lying below the center of buoyancy.This way, rotation of the vehicle from its resting state causes a corrective “restoring”moment to be applied to the vehicle. As derived in more detail in [10], these resultingforces and moments can be expressed as𝑋𝑅 (𝑊 𝐵)sin (𝜃)(8a)𝑌𝑅 (𝑊 𝐵)cos (𝜃)sin (𝜙)(8b)𝑍𝑅 (𝑊 𝐵)cos (𝜃)cos (𝜙)(8c)𝐾𝑅 (𝑦𝐺 𝑊 𝑦𝐵 𝐵) cos(𝜃) cos(𝜙) (𝑧𝐺 𝑊 𝑧𝐵 𝐵) cos(𝜃) sin(𝜙)(8d)𝑀𝑅 (𝑥𝐺 𝑊 𝑥𝐵 𝐵) cos(𝜃) cos(𝜙) (𝑧𝐺 𝑊 𝑧𝐵 𝐵) sin(𝜃)(8e)𝑁𝑅 (𝑥𝐺 𝑊 𝑥𝐵 𝐵) cos(𝜃) sin(𝜙) (𝑦𝐺 𝑊 𝑦𝐵 𝐵) sin(𝜃)(8f)where W represents the dry weight of the AUV, and B is the buoyancy force.2.5.4. Added MassAs a rigid body moves through a fluid, there are pressure-induced forces separatefrom drag associated with how the body is required to accelerate the surrounding fluidduring unsteady motion. This is called “added mass” and is a function of the geometry ofthe vehicle. Though these parameters are typically ignored in aerial applications due tothe low density of air, they must be accounted for in underwater analyses since the14

density of the fluid is much higher and on the same order as that of the rigid body. Theforces and moments are expressed as described in [13]𝐹𝑗 𝑈̇𝑖 𝑚𝑖𝑗 𝜀𝑗𝑘𝑙 𝑈𝑖 𝜔𝑘 𝑚𝑙𝑖(9a)𝑀𝑗 𝑈̇𝑖 𝑚𝑗 3,𝑖 𝜀𝑗𝑘𝑙 𝑈𝑖 𝜔𝑘 𝑚𝑙 3,𝑖 𝜀𝑗𝑘𝑙 𝑈𝑘 𝑈𝑖 𝑚𝑙𝑖(9b)where j denotes the direction of the force, i 1, 2, 3, 4, 5, 6 and j, k, l 1, 2, 3. εjkl is thealternating tensor where𝜀𝑗𝑘𝑙0; if any pair of the indices j, k, l are equalif j, k, l are in cyclic order { 1; 1;if j, k, l are in anti cyclic order(10)The overall effect of this added mass is more commonly condensed into ansymmetrical added mass or inertia matrix using Society of Naval Architects and MarineEngineers (SNAME) notation [14] as𝑀𝐴 𝑀𝑢̇[ 𝑀𝑤̇𝑁𝑤̇𝑋𝑝̇ 𝑋𝑞̇𝑌𝑝̇ 𝑌𝑞̇𝑍𝑝̇ 𝑍𝑞̇𝐾𝑝̇ 𝐾𝑞̇𝑀𝑝̇ 𝑀𝑞̇𝑁𝑝̇ 𝑀𝑟̇𝑁𝑟̇ ](11)Overall, the expanded equations for forces and moments on the rigid body due tothe added mass terms can be expressed as derived by Imlay [15],𝑋𝐴 𝑋𝑢̇ 𝑢̇ 𝑋𝑤̇ (𝑤̇ 𝑢𝑞) 𝑋𝑞̇ 𝑞̇ 𝑍𝑤̇ 𝑤𝑞 𝑍𝑞̇ 𝑞 2 𝑋𝑣̇ 𝑣̇ 𝑋𝑝̇ 𝑝̇ 𝑋𝑟̇ 𝑟̇ 𝑌𝑣̇ 𝑣𝑟 𝑌𝑝̇ 𝑟𝑝 𝑌𝑟̇ 𝑟 2 𝑋𝑣̇ 𝑢𝑟 𝑌𝑤̇ 𝑤𝑟 𝑌𝑤̇ 𝑣𝑞 𝑍𝑝̇ 𝑝𝑞 (𝑌𝑞̇ 𝑍𝑟̇ )𝑞𝑟(12a)15

𝑌𝐴 𝑋𝑣̇ 𝑢̇ 𝑌𝑤̇ 𝑤̇ 𝑌𝑞̇ 𝑞̇ 𝑌𝑣̇ 𝑣̇ 𝑌𝑝̇ 𝑝̇ 𝑌𝑟̇ 𝑟̇ 𝑋𝑣̇ 𝑣𝑟 𝑌𝑤̇ 𝑣𝑝 𝑋𝑟̇ 𝑟 2 (𝑋𝑝̇ 𝑍𝑟̇ )𝑟𝑝 𝑍𝑝̇ 𝑝2 𝑋𝑤̇ (𝑢𝑝 𝑤𝑟) 𝑋𝑢̇ 𝑢𝑟 𝑍𝑤̇ 𝑤𝑝 𝑍𝑞̇ 𝑝𝑞 𝑋𝑞̇ 𝑞𝑟(12b)𝑍𝐴 𝑋𝑤̇ (𝑢̇ 𝑤𝑞) 𝑍𝑤̇ 𝑤̇ 𝑍𝑞̇ 𝑞̇ 𝑋𝑢̇ 𝑢𝑞 𝑋𝑞̇ 𝑞 2 𝑌𝑤̇ 𝑣̇ 𝑍𝑝̇ 𝑝̇ 𝑍𝑟̇ 𝑟̇ 𝑌𝑣̇ 𝑣𝑝 𝑌𝑟̇ 𝑟𝑝 𝑌𝑝̇ 𝑝2 𝑋𝑣̇ 𝑢𝑝 𝑌𝑤̇ 𝑤𝑝 𝑋𝑣̇ 𝑣𝑞 (𝑋𝑝̇ 𝑌𝑞̇ )𝑝𝑞 𝑋𝑟̇ 𝑞𝑟(12c)𝐾𝐴 𝑋𝑝̇ 𝑢̇ 𝑍𝑝̇ 𝑤̇ 𝐾𝑞̇ 𝑞̇ 𝑋𝑣̇ 𝑤𝑢 𝑋𝑟̇ 𝑢𝑞 𝑌𝑤̇ 𝑤 2 (𝑌𝑞̇ 𝑍𝑟̇ )𝑤𝑞 𝑀𝑟̇ 𝑞 2 𝑌𝑝̇ 𝑣̇ 𝐾𝑝̇ 𝑝̇ 𝐾𝑟̇ 𝑟̇ 𝑌𝑤̇ 𝑣 2 (𝑌𝑞̇ 𝑍𝑟̇ )𝑣𝑟 𝑍𝑝̇ 𝑣𝑝 𝑀𝑟̇ 𝑟 2 𝐾𝑞̇ 𝑟𝑝 𝑋𝑤̇ 𝑢𝑣 (𝑌𝑣̇ 𝑍𝑤̇ )𝑣𝑤 (𝑌𝑟̇ 𝑍𝑞̇ )𝑤𝑟 𝑌𝑝̇ 𝑤𝑝 𝑋𝑞̇ 𝑢𝑟 (𝑌𝑟̇ 𝑍𝑞̇ )𝑣𝑞 𝐾𝑟̇ 𝑝𝑞 (𝑀𝑞̇ 𝑁𝑟̇ )𝑞𝑟(12d)𝑀𝐴 𝑋𝑞̇ (𝑢̇ 𝑤𝑞) 𝑍𝑞̇ (𝑤̇ 𝑢𝑞) 𝑀𝑞̇ 𝑞̇ 𝑋𝑤̇ (𝑢2 𝑤 2 ) (𝑍𝑤̇ 𝑋𝑢̇ )𝑤𝑢 𝑌𝑞̇ 𝑣̇ 𝐾𝑞̇ 𝑝̇ 𝑀𝑟̇ 𝑟̇ 𝑌𝑝̇ 𝑣𝑟 𝑌𝑟̇ 𝑣𝑝 𝐾𝑟̇ (𝑝2 𝑟 2 ) (𝐾𝑝̇ 𝑁𝑟̇ )𝑟𝑝 𝑌𝑤̇ 𝑢𝑣 𝑋𝑣̇ 𝑣𝑤 (𝑋𝑟̇ 𝑍𝑝̇ )(𝑢𝑝 𝑤𝑟) (𝑋𝑝̇ 𝑍𝑟̇ )(𝑤𝑝 𝑢𝑟) 𝑀𝑟̇ 𝑝𝑞 𝐾𝑞̇ 𝑞𝑟(12e)𝑁𝐴 𝑋𝑟̇ 𝑢̇ 𝑍𝑟̇ 𝑤̇ 𝑀𝑟̇ 𝑞̇ 𝑋𝑣̇ 𝑢2 𝑌𝑤̇ 𝑤𝑢 (𝑋𝑝̇ 𝑌𝑞̇ )𝑢𝑞 𝑍𝑝̇ 𝑤𝑞

Design (CAD) model in SolidWorks. This underwater vehicle seeks to assist personnel in numerous underwater applications in a more cost-effective manner than previous robots. Figure 1: LoCO in untethered deployment in the Caribbean Sea near Barbados [4]. Figure 2: LoCO CAD model in SolidWorks.

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